Rubber Band Area

Rubber Band Area

Below there are 5 points with a tight rubber band wrapped around all of them, creating a shape.  4 of the points are arranged in a square.  1 of the points is rotating around the center of the square, as shown.  The amount of rotation can be controlled using the slider.

GOAL: Write a function that takes the amount of rotation as input and outputs the area of the space inside the rubber band.

Extensions

  1. Is the function continuous?  Is it differentiable?
  2. When is the perimeter changing the fastest?
  3. All 4 corners of the square can be moved.  Can you generalize your function to work for squares of any side length?
  4. Move the corners to create quadrilaterals other than squares and try to write functions for those.
  5. What other questions can you ask about this situation?